Open Chaotic Cavities Research Articles

Microwave experiments using open chaotic cavities in the realm of the effective Hamiltonian formalism

AbstractThe description of open quantum system successfully uses the concept of the effective Hamiltonian, which takes into account the coupling to the environment. The concept can be extended to classical waves. Experimentally, classical waves are very convenient as they allow to control boundaries, coupling and absorption. We combine here recent works with microwaves investigating properties of open systems which can be described by an effective Hamiltonian. This ranges from spectral properties like modifications to level dynamics, width distribution and coupling fidelity to spatial properties like intensity distributions and complexness parameter which describes the non‐orthogonality of eigenfunctions.

Chaotic escape from an open vase-shaped cavity. I. Numerical and experimental results

We present part I in a two-part study of an open chaotic cavity shaped as a vase. The vase possesses an unstable periodic orbit in its neck. Trajectories passing through this orbit escape without return. For our analysis, we consider a family of trajectories launched from a point on the vase boundary. We imagine a vertical array of detectors past the unstable periodic orbit and, for each escaping trajectory, record the propagation time and the vertical detector position. We find that the escape time exhibits a complicated recursive structure. This recursive structure is explored in part I of our study. We present an approximation to the Helmholtz equation for waves escaping the vase. By choosing a set of detector points, we interpolate trajectories connecting the source to the different detector points. We use these interpolated classical trajectories to construct the solution to the wave equation at a detector point. Finally, we construct a plot of the detector position versus the escape time and compare this graph to the results of an experiment using classical ultrasound waves. We find that generally the classical trajectories organize the escaping ultrasound waves.

Largest Schmidt eigenvalue of random pure states and conductance distribution in chaoticcavities

A strategy to evaluate the distribution of the largest Schmidt eigenvalue forrandom pure states of bipartite systems is proposed. We point out thatthe multiple integral defining the sought quantity for a bipartition of sizesN, M is formally identical (upon simple algebraic manipulations) to the one providing theprobability density of Landauer conductance in open chaotic cavities supportingN andM electronic channels in the two leads. Known results about the latter can then bestraightforwardly employed in the former problem for systems with both broken (β = 2) andpreserved (β = 1) time-reversal symmetry. The analytical results, yielding a continuous but noteverywhere analytic distribution, are in excellent agreement with numerical simulations.

Statistics of resonance states in a weakly open chaotic cavity with continuously distributed losses

In this Rapid Communication, we demonstrate that a non-Hermitian random matrix description can account for both spectral and spatial statistics of resonance states in a weakly open chaotic wave system with continuously distributed losses. More specifically, the statistics of resonance states in an open two-dimensional chaotic microwave cavity are investigated by solving the Maxwell equations with lossy boundaries subject to Ohmic dissipation. We successfully compare the statistics of its complex-valued resonance states and associated widths with analytical predictions based on a non-Hermitian effective Hamiltonian model defined by a finite number of fictitious open channels.

Chaotic microcavity laser with high quality factor and unidirectional output

Here we report lasing action in lima\c{c}on-shaped GaAs microdisks with quantum dots (QDs) embedded. Although the intracavity ray dynamics is predominantly chaotic, high-$Q$ modes are concentrated in the region $\chi > \chi_c$ as a result of wave localization. Strong optical confinement by total internal reflection leads to very low lasing threshold. Our measurements show that all the lasing modes have output in the same direction, regardless of their wavelengths and intracavity mode structures. This universal emission direction is determined by directed phase space flow of optical rays in the open chaotic cavity. The divergence angle of output beam is less than 40 degree. The unidirectionality proves to be robust against small deviations of the real cavity shape and size from the designed values.

Distribution of resonances in the quantum open baker map

We study relevant features of the spectrum of the quantum open baker map. The opening consists of a cut along the momentum p direction of the 2-torus phase space, modeling an open chaotic cavity. We study briefly the classical forward trapped set and analyze the corresponding quantum nonunitary evolution operator. The distribution of eigenvalues depends strongly on the location of the escape region with respect to the central discontinuity of this map. This introduces new ingredients to the association among the classical escape and quantum decay rates. Finally, we could verify that the validity of the fractal Weyl law holds in all cases.

Spacing statistics in two-mode random lasing

The distribution of spacings between the lasing frequencies for an ensemble of random lasers in the two-mode regime was computed. The random lasers are implemented as open chaotic cavities filled with an active medium. The spectral properties of the passive cavities are modeled with non-Hermitian random matrices. The spacing distribution is found to depend on the relation between the gain-profile width and the mean spacing of the passive-cavity modes. The distribution displays mode repulsion and, under certain conditions, agrees with the Wigner surmise. The role of mode competition is discussed.

Statistical study of the conductance and shot noise in open quantum-chaotic cavities: Contribution from whispering gallery modes

In the past, a maximum-entropy model was introduced and applied to the study of statistical scattering by chaotic cavities, when short paths may play an important role in the scattering process. In particular, the validity of the model was investigated in relation with the statistical properties of the conductance in open chaotic cavities. In this article we investigate further the validity of the maximum-entropy model, by comparing the theoretical predictions with the results of computer simulations, in which the Schroedinger equation is solved numerically inside the cavity for one and two open channels in the leads; we analyze, in addition to the conductance, the zero-frequency limit of the shot-noise power spectrum. We also obtain theoretical results for the ensemble average of this last quantity, for the orthogonal and unitary cases of the circular ensemble and an arbitrary number of channels. Generally speaking, the agreement between theory and numerics is good. In some of the cavities that we study, short paths consist of whispering gallery modes, which were excluded in previous studies. These cavities turn out to be all the more interesting, as it is in relation with them that we found certain systematic discrepancies in the comparison with theory. We give evidence that it is the lack of stationarity inside the energy interval that is analyzed, and hence the lack of ergodicity that gives rise to the discrepancies. Indeed, the agreement between theory and numerical simulations is improved when the energy interval is reduced to a point and the statistics is then collected over an ensemble. It thus appears that the maximum-entropy model is valid beyond the domain where it was originally derived. An understanding of this situation is still lacking at the present moment.

Charge fluctuations in open chaotic cavities

We present a discussion of the charge response and the charge fluctuations of mesoscopic chaotic cavities in terms of a generalized Wigner-Smith matrix. The Wigner-Smith matrix is well known in investigations of time-delay of quantum scattering. It is expressed in terms of the scattering matrix and its derivatives with energy. We consider a similar matrix but instead of an energy derivative we investigate the derivative with regard to the electric potential. The resulting matrix is then the operator of charge. If this charge operator is combined with a self-consistent treatment of Coulomb interaction, the charge operator determines the capacitance of the system, the non-dissipative ac-linear response, the RC-time with a novel charge relaxation resistance, and in the presence of transport a resistance that governs the displacement currents induced into a nearby conductor. In particular these capacitances and resistances determine the relaxation rate and dephasing rate of a nearby qubit (a double quantum dot). We discuss the role of screening of mesoscopic chaotic detectors. Coulomb interaction effects in quantum pumping and in photon assisted electron-hole shot noise are treated similarly. For the latter we present novel results for chaotic cavities with non-ideal leads.

Measurement of Long-Range Wave-Function Correlations in an Open Microwave Billiard

We investigate the statistical properties of wave functions in an open chaotic cavity. When the number of channels in the openings of the billiard is increased by varying the frequency, wave functions cross over from real to complex. The distribution of the phase rigidity, which characterizes the degree to which a wave function is complex, and long-range correlations of intensity and current density are studied as a function of the number of channels in the openings. All measured quantities are in perfect agreement with theoretical predictions.

Efficiency of mesoscopic detectors.

We consider a mesoscopic measuring device whose conductance is sensitive to the state of a two-level system. The detector is described with the help of its scattering matrix. Its elements can be used to calculate the relaxation and decoherence times of the system, and determine the characteristic time for a reliable measurement. We derive conditions needed for an efficient ratio of decoherence and measurement times. To illustrate the theory we discuss the distribution function of the efficiency of an ensemble of open chaotic cavities.

Semiclassical theory of conductance and noise in open chaotic cavities

Conductance and shot noise of an open cavity with diffusive boundary scattering are calculated within the Boltzmann-Langevin approach. In particular, conductance contains a nonuniversal geometric contribution, originating from the presence of open contacts. Subsequently, universal expressions for multiterminal conductance and noise, valid for all chaotic cavities, are obtained classically, based on the fact that the distribution function in the cavity depends only on energy, and using the principle of minimal correlations.